Canon law provides stringent safeguards protecting the rights of spouses to be notified of annulment proceedings and inviting both spouses to participate if they wish. Indeed, if the spouses are not notified, it is possible grounds for declaring the entire procedure null and void (cf. canon 1511). It is a matter of justice to both spouses to give them proper notice and the ability to respond to a case against their marriage.
That said, in extremely exceptional cases it is possible that the tribunal could declare that due to moral impossibility one of the spouses cannot be notified. Your friend should make the situation known to the tribunal at the time he files for annulment and be prepared to back up his claim with documentation (e.g., police reports, medical records, court papers). The tribunal will not simply accept his word on the matter that his ex-wife is "mentally unstable"; the tribunal will have to verify the claim.
Even if the tribunal should make the relatively rare exception in your friend's case, his ex-wife may well find out about the annulment because it is a matter of public record. For example, should an annulment be declared, it will be documented on both spouses' sacramental records in the parishes of their baptism. Should she order a copy of her sacramental record, she will know that an annulment took place without her knowledge. It is very possible that a secret annulment may unhinge such a person even more
than if the annulment proceedings had taken place with her knowledge and with the possibility of her input. If your friend can take absolutely no risk of his ex-wife finding out about the proceedings, it may well be better to put off the proceedings to a more opportune time. This would mean that he could not marry you and should not engage in a romantic relationship with you; but, if his safety is at stake, then his safety concerns must override your marriage plans.
If she is notified, your friend's ex-wife will be invited to participate in the proceedings. The tribunal can proceed even over her objections, but the tribunal will allow her to defend the validity of her marriage should she wish to do so. If she is as unbalanced as you claim, that fact likely will be apparent to the tribunal and will be taken into consideration. Your friend can substantiate his ex-wife's illness through documentation from police, lawyers, doctors, judges, etc. If the tribunal determines that the illness was present from the beginning of the marriage, it may in itself be grounds for the annulment because such an illness can impair the ability to give the required matrimonial consent.
In the meantime, while these proceedings are progressing, my advice to you is to end your romantic involvement with your friend. Unless and until the tribunal declares his first marriage null, the Church presumes your friend to be a married man and expects him to order his relationships according to that presumption. Bluntly, that means that a married man cannot have a girlfriend. You are certainly free to remain platonic friends; but you should seriously consider ending any romantic relationship.
Annulments and the Catholic Church
by Edward Peters